Suslin’s algorithms for reduction of unimodular rows

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Abstract

A well-known lemma of Suslin says that for a commutative ring A if (v1(X),,vn(X))(A[X])n is unimodular where v1 is monic and n3, then there exist γ1,,γEn1(A[X]) such that the ideal generated by Res(v1,e1.γ1t(v2,,vn)),,Res(v1,e1.γt(v2,,vn)) equals A. This lemma played a central role in the resolution of Serre’s Conjecture. In the case where A contains a set E of cardinality greater than degv1+1 such that yy is invertible for each yy in E, we prove that the γi can simply correspond to the elementary operations L1L1+yij=2n1uj+1Lj, 1i=degv1+1, where u1v1++unvn=1. These efficient elementary operations enable us to give new and simple algorithms for reducing unimodular rows with entries in K[X1,,Xk] to t(1,0,,0) using elementary operations in the case where K is an infinite field. Another feature of this paper is that it shows that the concrete local–global principles can produce competitive complexity bounds.

MSC

13C10
19A13
14Q20
03F65

Keywords

Quillen–Suslin theorem
Suslin’s stability theorem
Constructive mathematics
Computer algebra

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